… … Hence, in the long term, there will be 50 absent and 150 present on each day.

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Finding the Steady State without a Calculator

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Example 1 (continued)

Find the number of students present/absent on each day in the long term

Solution

… … Let P = the number present in the steady state

… … Let A = the number absent in the steady state
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… … Total Students = $200$,
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… … so $P + A = 200$
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… … so $A = 200 - P$ … {Equation 1}

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… … Express the rule for P(next) in terms of P and A
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… … $P(next) = 0.9 \times P + 0.3 \times A$
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… … but in the steady state, $P(next) = P(previous)$
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… … so
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… … $P = 0.9P + 0.3A$ … {Equation 2}

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… … {Substitute Eqn 1 into Eqn 2}
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… … $P = 0.9P + 0.3(200 - P)$
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… … $P = 0.9P + 60 – 0.3P$
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… … $P = 0.6P + 60$
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… … $0.4P = 60$
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… … $P = 150$
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… … {Using Eqn 1}
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… … $A = 200 - P$
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… … $A = 50$
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… … So in the long term , on any given day

… … … 150 students will be present
… … … 50 students will be absent

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Example 2

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Commuters travelling into the centre of Trenchtown use either the bus or the train. Research shows that each month, 20% of those using the bus switch to train travel and 30% of those using the train switch to bus travel.

At the start of January, 4800 were using the bus and 3600 were using the train.